Spectral Analysis of N-Body Schrödinger Operators at Two-Cluster Thresholds

This book provides a systematic study of spectral and scattering theory nbsp;for many-body Schrödinger operators at two-cluster thresholds. While the nbsp;two-body problem (reduced after separation of the centre of mass motion to a one-body problem at zero energy) is a well-studied subject, the nbsp;literature on nbsp;many-body threshold problems nbsp;is sparse. However, the authors' analysis covers for example the system of three particles nbsp;interacting by Coulomb potentials and restricted to a small energy nbsp;region to the right of a fixed nonzero two-body eigenvalue. In general, nbsp;the authors address the question: How do scattering quantities for the nbsp;many-body atomic and molecular models behave within the limit when the nbsp;total energy approaches a fixed two-cluster threshold? This includes nbsp;mapping properties and singularities of the limiting scattering matrix, nbsp;asymptotics of the total scattering cross section, and absence of nbsp;transmission from one channel to another in the small inter-cluster nbsp;kinetic energy region. The authors' principal tools are the nbsp;Feshbach-Grushin dimension reduction method and spectral analysis based nbsp;on a certain Mourre estimate. Additional topics of independent interest nbsp;are the limiting absorption principle, micro-local resolvent estimates, nbsp;Rellich- and Sommerfeld-type theorems and asymptotics of the limiting nbsp;resolvents at thresholds. The mathematical physics field under study is nbsp;very rich, and there are many open problems, several of them stated nbsp;explicitly in the book for the interested reader.nbsp;